Investigation of nodal domains in the chaotic microwave ray-splitting rough billiard

TL;DR

This study experimentally investigates the properties of nodal domains in chaotic microwave billiards, confirming theoretical predictions about their distribution, number, and scaling behavior in the ergodic regime.

Contribution

First experimental measurement of nodal domains in chaotic microwave ray-splitting billiards, validating theoretical models of their statistical properties and distributions.

Findings

Number of nodal domains per level approaches theoretical value as N increases.

Distribution of nodal domain areas follows a power law with exponent ~2.14.

Wave functions are Gaussian distributed and extended over the energy surface in the ergodic regime.

Abstract

We study experimentally nodal domains of wave functions (electric field distributions) lying in the regime of Shnirelman ergodicity in the chaotic microwave half-circular ray-splitting rough billiard. For this aim the wave functions Psi_N of the billiard were measured up to the level number N=415. We show that in the regime of Shnirelman ergodicity (N>208) wave functions of the chaotic half-circular microwave ray-splitting rough billiard are extended over the whole energy surface and the amplitude distributions are Gaussian. For such ergodic wave functions the dependence of the number of nodal domains aleph_N on the level number N was found. We show that in the limit N->infty the least squares fit of the experimental data yields aleph_N/N = 0.063 +- 0.023 that is close to the theoretical prediction aleph_N/N = 0.062. We demonstrate that for higher level numbers N = 215-415 the variance of the mean number of nodal domains sigma^2_N/ N is scattered around the theoretical limit sigma^2_N /N = 0.05. We also found that the distribution of the areas s of nodal domains has power behavior n_s ~ s^{-tau}, where the scaling exponent is equal to tau = 2.14 +- 0.12. This result is in a good agreement with the prediction of percolation theory.